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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    <a name="line.17"></a>
<FONT color="green">018</FONT>    package org.apache.commons.math3.linear;<a name="line.18"></a>
<FONT color="green">019</FONT>    <a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.complex.Complex;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.MathArithmeticException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.MathUnsupportedOperationException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.exception.MaxCountExceededException;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.exception.DimensionMismatchException;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.25"></a>
<FONT color="green">026</FONT>    import org.apache.commons.math3.util.Precision;<a name="line.26"></a>
<FONT color="green">027</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.27"></a>
<FONT color="green">028</FONT>    <a name="line.28"></a>
<FONT color="green">029</FONT>    /**<a name="line.29"></a>
<FONT color="green">030</FONT>     * Calculates the eigen decomposition of a real matrix.<a name="line.30"></a>
<FONT color="green">031</FONT>     * &lt;p&gt;The eigen decomposition of matrix A is a set of two matrices:<a name="line.31"></a>
<FONT color="green">032</FONT>     * V and D such that A = V &amp;times; D &amp;times; V&lt;sup&gt;T&lt;/sup&gt;.<a name="line.32"></a>
<FONT color="green">033</FONT>     * A, V and D are all m &amp;times; m matrices.&lt;/p&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     * &lt;p&gt;This class is similar in spirit to the &lt;code&gt;EigenvalueDecomposition&lt;/code&gt;<a name="line.34"></a>
<FONT color="green">035</FONT>     * class from the &lt;a href="http://math.nist.gov/javanumerics/jama/"&gt;JAMA&lt;/a&gt;<a name="line.35"></a>
<FONT color="green">036</FONT>     * library, with the following changes:&lt;/p&gt;<a name="line.36"></a>
<FONT color="green">037</FONT>     * &lt;ul&gt;<a name="line.37"></a>
<FONT color="green">038</FONT>     *   &lt;li&gt;a {@link #getVT() getVt} method has been added,&lt;/li&gt;<a name="line.38"></a>
<FONT color="green">039</FONT>     *   &lt;li&gt;two {@link #getRealEigenvalue(int) getRealEigenvalue} and {@link #getImagEigenvalue(int)<a name="line.39"></a>
<FONT color="green">040</FONT>     *   getImagEigenvalue} methods to pick up a single eigenvalue have been added,&lt;/li&gt;<a name="line.40"></a>
<FONT color="green">041</FONT>     *   &lt;li&gt;a {@link #getEigenvector(int) getEigenvector} method to pick up a single<a name="line.41"></a>
<FONT color="green">042</FONT>     *   eigenvector has been added,&lt;/li&gt;<a name="line.42"></a>
<FONT color="green">043</FONT>     *   &lt;li&gt;a {@link #getDeterminant() getDeterminant} method has been added.&lt;/li&gt;<a name="line.43"></a>
<FONT color="green">044</FONT>     *   &lt;li&gt;a {@link #getSolver() getSolver} method has been added.&lt;/li&gt;<a name="line.44"></a>
<FONT color="green">045</FONT>     * &lt;/ul&gt;<a name="line.45"></a>
<FONT color="green">046</FONT>     * &lt;p&gt;<a name="line.46"></a>
<FONT color="green">047</FONT>     * As of 3.1, this class supports general real matrices (both symmetric and non-symmetric):<a name="line.47"></a>
<FONT color="green">048</FONT>     * &lt;/p&gt;<a name="line.48"></a>
<FONT color="green">049</FONT>     * &lt;p&gt;<a name="line.49"></a>
<FONT color="green">050</FONT>     * If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector<a name="line.50"></a>
<FONT color="green">051</FONT>     * matrix V is orthogonal, i.e. A = V.multiply(D.multiply(V.transpose())) and<a name="line.51"></a>
<FONT color="green">052</FONT>     * V.multiply(V.transpose()) equals the identity matrix.<a name="line.52"></a>
<FONT color="green">053</FONT>     * &lt;/p&gt;<a name="line.53"></a>
<FONT color="green">054</FONT>     * &lt;p&gt;<a name="line.54"></a>
<FONT color="green">055</FONT>     * If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues<a name="line.55"></a>
<FONT color="green">056</FONT>     * in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks:<a name="line.56"></a>
<FONT color="green">057</FONT>     * &lt;pre&gt;<a name="line.57"></a>
<FONT color="green">058</FONT>     *    [lambda, mu    ]<a name="line.58"></a>
<FONT color="green">059</FONT>     *    [   -mu, lambda]<a name="line.59"></a>
<FONT color="green">060</FONT>     * &lt;/pre&gt;<a name="line.60"></a>
<FONT color="green">061</FONT>     * The columns of V represent the eigenvectors in the sense that A*V = V*D,<a name="line.61"></a>
<FONT color="green">062</FONT>     * i.e. A.multiply(V) equals V.multiply(D).<a name="line.62"></a>
<FONT color="green">063</FONT>     * The matrix V may be badly conditioned, or even singular, so the validity of the equation<a name="line.63"></a>
<FONT color="green">064</FONT>     * A = V*D*inverse(V) depends upon the condition of V.<a name="line.64"></a>
<FONT color="green">065</FONT>     * &lt;/p&gt;<a name="line.65"></a>
<FONT color="green">066</FONT>     * &lt;p&gt;<a name="line.66"></a>
<FONT color="green">067</FONT>     * This implementation is based on the paper by A. Drubrulle, R.S. Martin and<a name="line.67"></a>
<FONT color="green">068</FONT>     * J.H. Wilkinson "The Implicit QL Algorithm" in Wilksinson and Reinsch (1971)<a name="line.68"></a>
<FONT color="green">069</FONT>     * Handbook for automatic computation, vol. 2, Linear algebra, Springer-Verlag,<a name="line.69"></a>
<FONT color="green">070</FONT>     * New-York<a name="line.70"></a>
<FONT color="green">071</FONT>     * &lt;/p&gt;<a name="line.71"></a>
<FONT color="green">072</FONT>     * @see &lt;a href="http://mathworld.wolfram.com/EigenDecomposition.html"&gt;MathWorld&lt;/a&gt;<a name="line.72"></a>
<FONT color="green">073</FONT>     * @see &lt;a href="http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix"&gt;Wikipedia&lt;/a&gt;<a name="line.73"></a>
<FONT color="green">074</FONT>     * @version $Id: EigenDecomposition.java 1422195 2012-12-15 06:45:18Z psteitz $<a name="line.74"></a>
<FONT color="green">075</FONT>     * @since 2.0 (changed to concrete class in 3.0)<a name="line.75"></a>
<FONT color="green">076</FONT>     */<a name="line.76"></a>
<FONT color="green">077</FONT>    public class EigenDecomposition {<a name="line.77"></a>
<FONT color="green">078</FONT>        /** Internally used epsilon criteria. */<a name="line.78"></a>
<FONT color="green">079</FONT>        private static final double EPSILON = 1e-12;<a name="line.79"></a>
<FONT color="green">080</FONT>        /** Maximum number of iterations accepted in the implicit QL transformation */<a name="line.80"></a>
<FONT color="green">081</FONT>        private byte maxIter = 30;<a name="line.81"></a>
<FONT color="green">082</FONT>        /** Main diagonal of the tridiagonal matrix. */<a name="line.82"></a>
<FONT color="green">083</FONT>        private double[] main;<a name="line.83"></a>
<FONT color="green">084</FONT>        /** Secondary diagonal of the tridiagonal matrix. */<a name="line.84"></a>
<FONT color="green">085</FONT>        private double[] secondary;<a name="line.85"></a>
<FONT color="green">086</FONT>        /**<a name="line.86"></a>
<FONT color="green">087</FONT>         * Transformer to tridiagonal (may be null if matrix is already<a name="line.87"></a>
<FONT color="green">088</FONT>         * tridiagonal).<a name="line.88"></a>
<FONT color="green">089</FONT>         */<a name="line.89"></a>
<FONT color="green">090</FONT>        private TriDiagonalTransformer transformer;<a name="line.90"></a>
<FONT color="green">091</FONT>        /** Real part of the realEigenvalues. */<a name="line.91"></a>
<FONT color="green">092</FONT>        private double[] realEigenvalues;<a name="line.92"></a>
<FONT color="green">093</FONT>        /** Imaginary part of the realEigenvalues. */<a name="line.93"></a>
<FONT color="green">094</FONT>        private double[] imagEigenvalues;<a name="line.94"></a>
<FONT color="green">095</FONT>        /** Eigenvectors. */<a name="line.95"></a>
<FONT color="green">096</FONT>        private ArrayRealVector[] eigenvectors;<a name="line.96"></a>
<FONT color="green">097</FONT>        /** Cached value of V. */<a name="line.97"></a>
<FONT color="green">098</FONT>        private RealMatrix cachedV;<a name="line.98"></a>
<FONT color="green">099</FONT>        /** Cached value of D. */<a name="line.99"></a>
<FONT color="green">100</FONT>        private RealMatrix cachedD;<a name="line.100"></a>
<FONT color="green">101</FONT>        /** Cached value of Vt. */<a name="line.101"></a>
<FONT color="green">102</FONT>        private RealMatrix cachedVt;<a name="line.102"></a>
<FONT color="green">103</FONT>        /** Whether the matrix is symmetric. */<a name="line.103"></a>
<FONT color="green">104</FONT>        private final boolean isSymmetric;<a name="line.104"></a>
<FONT color="green">105</FONT>    <a name="line.105"></a>
<FONT color="green">106</FONT>        /**<a name="line.106"></a>
<FONT color="green">107</FONT>         * Calculates the eigen decomposition of the given real matrix.<a name="line.107"></a>
<FONT color="green">108</FONT>         * &lt;p&gt;<a name="line.108"></a>
<FONT color="green">109</FONT>         * Supports decomposition of a general matrix since 3.1.<a name="line.109"></a>
<FONT color="green">110</FONT>         *<a name="line.110"></a>
<FONT color="green">111</FONT>         * @param matrix Matrix to decompose.<a name="line.111"></a>
<FONT color="green">112</FONT>         * @throws MaxCountExceededException if the algorithm fails to converge.<a name="line.112"></a>
<FONT color="green">113</FONT>         * @throws MathArithmeticException if the decomposition of a general matrix<a name="line.113"></a>
<FONT color="green">114</FONT>         * results in a matrix with zero norm<a name="line.114"></a>
<FONT color="green">115</FONT>         * @since 3.1<a name="line.115"></a>
<FONT color="green">116</FONT>         */<a name="line.116"></a>
<FONT color="green">117</FONT>        public EigenDecomposition(final RealMatrix matrix)<a name="line.117"></a>
<FONT color="green">118</FONT>            throws MathArithmeticException {<a name="line.118"></a>
<FONT color="green">119</FONT>            final double symTol = 10 * matrix.getRowDimension() * matrix.getColumnDimension() * Precision.EPSILON;<a name="line.119"></a>
<FONT color="green">120</FONT>            isSymmetric = MatrixUtils.isSymmetric(matrix, symTol);<a name="line.120"></a>
<FONT color="green">121</FONT>            if (isSymmetric) {<a name="line.121"></a>
<FONT color="green">122</FONT>                transformToTridiagonal(matrix);<a name="line.122"></a>
<FONT color="green">123</FONT>                findEigenVectors(transformer.getQ().getData());<a name="line.123"></a>
<FONT color="green">124</FONT>            } else {<a name="line.124"></a>
<FONT color="green">125</FONT>                final SchurTransformer t = transformToSchur(matrix);<a name="line.125"></a>
<FONT color="green">126</FONT>                findEigenVectorsFromSchur(t);<a name="line.126"></a>
<FONT color="green">127</FONT>            }<a name="line.127"></a>
<FONT color="green">128</FONT>        }<a name="line.128"></a>
<FONT color="green">129</FONT>    <a name="line.129"></a>
<FONT color="green">130</FONT>        /**<a name="line.130"></a>
<FONT color="green">131</FONT>         * Calculates the eigen decomposition of the given real matrix.<a name="line.131"></a>
<FONT color="green">132</FONT>         *<a name="line.132"></a>
<FONT color="green">133</FONT>         * @param matrix Matrix to decompose.<a name="line.133"></a>
<FONT color="green">134</FONT>         * @param splitTolerance Dummy parameter (present for backward<a name="line.134"></a>
<FONT color="green">135</FONT>         * compatibility only).<a name="line.135"></a>
<FONT color="green">136</FONT>         * @throws MathArithmeticException  if the decomposition of a general matrix<a name="line.136"></a>
<FONT color="green">137</FONT>         * results in a matrix with zero norm<a name="line.137"></a>
<FONT color="green">138</FONT>         * @throws MaxCountExceededException if the algorithm fails to converge.<a name="line.138"></a>
<FONT color="green">139</FONT>         * @deprecated in 3.1 (to be removed in 4.0) due to unused parameter<a name="line.139"></a>
<FONT color="green">140</FONT>         */<a name="line.140"></a>
<FONT color="green">141</FONT>        @Deprecated<a name="line.141"></a>
<FONT color="green">142</FONT>        public EigenDecomposition(final RealMatrix matrix,<a name="line.142"></a>
<FONT color="green">143</FONT>                                  final double splitTolerance)<a name="line.143"></a>
<FONT color="green">144</FONT>            throws MathArithmeticException {<a name="line.144"></a>
<FONT color="green">145</FONT>            this(matrix);<a name="line.145"></a>
<FONT color="green">146</FONT>        }<a name="line.146"></a>
<FONT color="green">147</FONT>    <a name="line.147"></a>
<FONT color="green">148</FONT>        /**<a name="line.148"></a>
<FONT color="green">149</FONT>         * Calculates the eigen decomposition of the symmetric tridiagonal<a name="line.149"></a>
<FONT color="green">150</FONT>         * matrix.  The Householder matrix is assumed to be the identity matrix.<a name="line.150"></a>
<FONT color="green">151</FONT>         *<a name="line.151"></a>
<FONT color="green">152</FONT>         * @param main Main diagonal of the symmetric tridiagonal form.<a name="line.152"></a>
<FONT color="green">153</FONT>         * @param secondary Secondary of the tridiagonal form.<a name="line.153"></a>
<FONT color="green">154</FONT>         * @throws MaxCountExceededException if the algorithm fails to converge.<a name="line.154"></a>
<FONT color="green">155</FONT>         * @since 3.1<a name="line.155"></a>
<FONT color="green">156</FONT>         */<a name="line.156"></a>
<FONT color="green">157</FONT>        public EigenDecomposition(final double[] main, final double[] secondary) {<a name="line.157"></a>
<FONT color="green">158</FONT>            isSymmetric = true;<a name="line.158"></a>
<FONT color="green">159</FONT>            this.main      = main.clone();<a name="line.159"></a>
<FONT color="green">160</FONT>            this.secondary = secondary.clone();<a name="line.160"></a>
<FONT color="green">161</FONT>            transformer    = null;<a name="line.161"></a>
<FONT color="green">162</FONT>            final int size = main.length;<a name="line.162"></a>
<FONT color="green">163</FONT>            final double[][] z = new double[size][size];<a name="line.163"></a>
<FONT color="green">164</FONT>            for (int i = 0; i &lt; size; i++) {<a name="line.164"></a>
<FONT color="green">165</FONT>                z[i][i] = 1.0;<a name="line.165"></a>
<FONT color="green">166</FONT>            }<a name="line.166"></a>
<FONT color="green">167</FONT>            findEigenVectors(z);<a name="line.167"></a>
<FONT color="green">168</FONT>        }<a name="line.168"></a>
<FONT color="green">169</FONT>    <a name="line.169"></a>
<FONT color="green">170</FONT>        /**<a name="line.170"></a>
<FONT color="green">171</FONT>         * Calculates the eigen decomposition of the symmetric tridiagonal<a name="line.171"></a>
<FONT color="green">172</FONT>         * matrix.  The Householder matrix is assumed to be the identity matrix.<a name="line.172"></a>
<FONT color="green">173</FONT>         *<a name="line.173"></a>
<FONT color="green">174</FONT>         * @param main Main diagonal of the symmetric tridiagonal form.<a name="line.174"></a>
<FONT color="green">175</FONT>         * @param secondary Secondary of the tridiagonal form.<a name="line.175"></a>
<FONT color="green">176</FONT>         * @param splitTolerance Dummy parameter (present for backward<a name="line.176"></a>
<FONT color="green">177</FONT>         * compatibility only).<a name="line.177"></a>
<FONT color="green">178</FONT>         * @throws MaxCountExceededException if the algorithm fails to converge.<a name="line.178"></a>
<FONT color="green">179</FONT>         * @deprecated in 3.1 (to be removed in 4.0) due to unused parameter<a name="line.179"></a>
<FONT color="green">180</FONT>         */<a name="line.180"></a>
<FONT color="green">181</FONT>        @Deprecated<a name="line.181"></a>
<FONT color="green">182</FONT>        public EigenDecomposition(final double[] main, final double[] secondary,<a name="line.182"></a>
<FONT color="green">183</FONT>                                  final double splitTolerance) {<a name="line.183"></a>
<FONT color="green">184</FONT>            this(main, secondary);<a name="line.184"></a>
<FONT color="green">185</FONT>        }<a name="line.185"></a>
<FONT color="green">186</FONT>    <a name="line.186"></a>
<FONT color="green">187</FONT>        /**<a name="line.187"></a>
<FONT color="green">188</FONT>         * Gets the matrix V of the decomposition.<a name="line.188"></a>
<FONT color="green">189</FONT>         * V is an orthogonal matrix, i.e. its transpose is also its inverse.<a name="line.189"></a>
<FONT color="green">190</FONT>         * The columns of V are the eigenvectors of the original matrix.<a name="line.190"></a>
<FONT color="green">191</FONT>         * No assumption is made about the orientation of the system axes formed<a name="line.191"></a>
<FONT color="green">192</FONT>         * by the columns of V (e.g. in a 3-dimension space, V can form a left-<a name="line.192"></a>
<FONT color="green">193</FONT>         * or right-handed system).<a name="line.193"></a>
<FONT color="green">194</FONT>         *<a name="line.194"></a>
<FONT color="green">195</FONT>         * @return the V matrix.<a name="line.195"></a>
<FONT color="green">196</FONT>         */<a name="line.196"></a>
<FONT color="green">197</FONT>        public RealMatrix getV() {<a name="line.197"></a>
<FONT color="green">198</FONT>    <a name="line.198"></a>
<FONT color="green">199</FONT>            if (cachedV == null) {<a name="line.199"></a>
<FONT color="green">200</FONT>                final int m = eigenvectors.length;<a name="line.200"></a>
<FONT color="green">201</FONT>                cachedV = MatrixUtils.createRealMatrix(m, m);<a name="line.201"></a>
<FONT color="green">202</FONT>                for (int k = 0; k &lt; m; ++k) {<a name="line.202"></a>
<FONT color="green">203</FONT>                    cachedV.setColumnVector(k, eigenvectors[k]);<a name="line.203"></a>
<FONT color="green">204</FONT>                }<a name="line.204"></a>
<FONT color="green">205</FONT>            }<a name="line.205"></a>
<FONT color="green">206</FONT>            // return the cached matrix<a name="line.206"></a>
<FONT color="green">207</FONT>            return cachedV;<a name="line.207"></a>
<FONT color="green">208</FONT>        }<a name="line.208"></a>
<FONT color="green">209</FONT>    <a name="line.209"></a>
<FONT color="green">210</FONT>        /**<a name="line.210"></a>
<FONT color="green">211</FONT>         * Gets the block diagonal matrix D of the decomposition.<a name="line.211"></a>
<FONT color="green">212</FONT>         * D is a block diagonal matrix.<a name="line.212"></a>
<FONT color="green">213</FONT>         * Real eigenvalues are on the diagonal while complex values are on<a name="line.213"></a>
<FONT color="green">214</FONT>         * 2x2 blocks { {real +imaginary}, {-imaginary, real} }.<a name="line.214"></a>
<FONT color="green">215</FONT>         *<a name="line.215"></a>
<FONT color="green">216</FONT>         * @return the D matrix.<a name="line.216"></a>
<FONT color="green">217</FONT>         *<a name="line.217"></a>
<FONT color="green">218</FONT>         * @see #getRealEigenvalues()<a name="line.218"></a>
<FONT color="green">219</FONT>         * @see #getImagEigenvalues()<a name="line.219"></a>
<FONT color="green">220</FONT>         */<a name="line.220"></a>
<FONT color="green">221</FONT>        public RealMatrix getD() {<a name="line.221"></a>
<FONT color="green">222</FONT>    <a name="line.222"></a>
<FONT color="green">223</FONT>            if (cachedD == null) {<a name="line.223"></a>
<FONT color="green">224</FONT>                // cache the matrix for subsequent calls<a name="line.224"></a>
<FONT color="green">225</FONT>                cachedD = MatrixUtils.createRealDiagonalMatrix(realEigenvalues);<a name="line.225"></a>
<FONT color="green">226</FONT>    <a name="line.226"></a>
<FONT color="green">227</FONT>                for (int i = 0; i &lt; imagEigenvalues.length; i++) {<a name="line.227"></a>
<FONT color="green">228</FONT>                    if (Precision.compareTo(imagEigenvalues[i], 0.0, EPSILON) &gt; 0) {<a name="line.228"></a>
<FONT color="green">229</FONT>                        cachedD.setEntry(i, i+1, imagEigenvalues[i]);<a name="line.229"></a>
<FONT color="green">230</FONT>                    } else if (Precision.compareTo(imagEigenvalues[i], 0.0, EPSILON) &lt; 0) {<a name="line.230"></a>
<FONT color="green">231</FONT>                        cachedD.setEntry(i, i-1, imagEigenvalues[i]);<a name="line.231"></a>
<FONT color="green">232</FONT>                    }<a name="line.232"></a>
<FONT color="green">233</FONT>                }<a name="line.233"></a>
<FONT color="green">234</FONT>            }<a name="line.234"></a>
<FONT color="green">235</FONT>            return cachedD;<a name="line.235"></a>
<FONT color="green">236</FONT>        }<a name="line.236"></a>
<FONT color="green">237</FONT>    <a name="line.237"></a>
<FONT color="green">238</FONT>        /**<a name="line.238"></a>
<FONT color="green">239</FONT>         * Gets the transpose of the matrix V of the decomposition.<a name="line.239"></a>
<FONT color="green">240</FONT>         * V is an orthogonal matrix, i.e. its transpose is also its inverse.<a name="line.240"></a>
<FONT color="green">241</FONT>         * The columns of V are the eigenvectors of the original matrix.<a name="line.241"></a>
<FONT color="green">242</FONT>         * No assumption is made about the orientation of the system axes formed<a name="line.242"></a>
<FONT color="green">243</FONT>         * by the columns of V (e.g. in a 3-dimension space, V can form a left-<a name="line.243"></a>
<FONT color="green">244</FONT>         * or right-handed system).<a name="line.244"></a>
<FONT color="green">245</FONT>         *<a name="line.245"></a>
<FONT color="green">246</FONT>         * @return the transpose of the V matrix.<a name="line.246"></a>
<FONT color="green">247</FONT>         */<a name="line.247"></a>
<FONT color="green">248</FONT>        public RealMatrix getVT() {<a name="line.248"></a>
<FONT color="green">249</FONT>    <a name="line.249"></a>
<FONT color="green">250</FONT>            if (cachedVt == null) {<a name="line.250"></a>
<FONT color="green">251</FONT>                final int m = eigenvectors.length;<a name="line.251"></a>
<FONT color="green">252</FONT>                cachedVt = MatrixUtils.createRealMatrix(m, m);<a name="line.252"></a>
<FONT color="green">253</FONT>                for (int k = 0; k &lt; m; ++k) {<a name="line.253"></a>
<FONT color="green">254</FONT>                    cachedVt.setRowVector(k, eigenvectors[k]);<a name="line.254"></a>
<FONT color="green">255</FONT>                }<a name="line.255"></a>
<FONT color="green">256</FONT>            }<a name="line.256"></a>
<FONT color="green">257</FONT>    <a name="line.257"></a>
<FONT color="green">258</FONT>            // return the cached matrix<a name="line.258"></a>
<FONT color="green">259</FONT>            return cachedVt;<a name="line.259"></a>
<FONT color="green">260</FONT>        }<a name="line.260"></a>
<FONT color="green">261</FONT>    <a name="line.261"></a>
<FONT color="green">262</FONT>        /**<a name="line.262"></a>
<FONT color="green">263</FONT>         * Returns whether the calculated eigen values are complex or real.<a name="line.263"></a>
<FONT color="green">264</FONT>         * &lt;p&gt;The method performs a zero check for each element of the<a name="line.264"></a>
<FONT color="green">265</FONT>         * {@link #getImagEigenvalues()} array and returns {@code true} if any<a name="line.265"></a>
<FONT color="green">266</FONT>         * element is not equal to zero.<a name="line.266"></a>
<FONT color="green">267</FONT>         *<a name="line.267"></a>
<FONT color="green">268</FONT>         * @return {@code true} if the eigen values are complex, {@code false} otherwise<a name="line.268"></a>
<FONT color="green">269</FONT>         * @since 3.1<a name="line.269"></a>
<FONT color="green">270</FONT>         */<a name="line.270"></a>
<FONT color="green">271</FONT>        public boolean hasComplexEigenvalues() {<a name="line.271"></a>
<FONT color="green">272</FONT>            for (int i = 0; i &lt; imagEigenvalues.length; i++) {<a name="line.272"></a>
<FONT color="green">273</FONT>                if (!Precision.equals(imagEigenvalues[i], 0.0, EPSILON)) {<a name="line.273"></a>
<FONT color="green">274</FONT>                    return true;<a name="line.274"></a>
<FONT color="green">275</FONT>                }<a name="line.275"></a>
<FONT color="green">276</FONT>            }<a name="line.276"></a>
<FONT color="green">277</FONT>            return false;<a name="line.277"></a>
<FONT color="green">278</FONT>        }<a name="line.278"></a>
<FONT color="green">279</FONT>    <a name="line.279"></a>
<FONT color="green">280</FONT>        /**<a name="line.280"></a>
<FONT color="green">281</FONT>         * Gets a copy of the real parts of the eigenvalues of the original matrix.<a name="line.281"></a>
<FONT color="green">282</FONT>         *<a name="line.282"></a>
<FONT color="green">283</FONT>         * @return a copy of the real parts of the eigenvalues of the original matrix.<a name="line.283"></a>
<FONT color="green">284</FONT>         *<a name="line.284"></a>
<FONT color="green">285</FONT>         * @see #getD()<a name="line.285"></a>
<FONT color="green">286</FONT>         * @see #getRealEigenvalue(int)<a name="line.286"></a>
<FONT color="green">287</FONT>         * @see #getImagEigenvalues()<a name="line.287"></a>
<FONT color="green">288</FONT>         */<a name="line.288"></a>
<FONT color="green">289</FONT>        public double[] getRealEigenvalues() {<a name="line.289"></a>
<FONT color="green">290</FONT>            return realEigenvalues.clone();<a name="line.290"></a>
<FONT color="green">291</FONT>        }<a name="line.291"></a>
<FONT color="green">292</FONT>    <a name="line.292"></a>
<FONT color="green">293</FONT>        /**<a name="line.293"></a>
<FONT color="green">294</FONT>         * Returns the real part of the i&lt;sup&gt;th&lt;/sup&gt; eigenvalue of the original<a name="line.294"></a>
<FONT color="green">295</FONT>         * matrix.<a name="line.295"></a>
<FONT color="green">296</FONT>         *<a name="line.296"></a>
<FONT color="green">297</FONT>         * @param i index of the eigenvalue (counting from 0)<a name="line.297"></a>
<FONT color="green">298</FONT>         * @return real part of the i&lt;sup&gt;th&lt;/sup&gt; eigenvalue of the original<a name="line.298"></a>
<FONT color="green">299</FONT>         * matrix.<a name="line.299"></a>
<FONT color="green">300</FONT>         *<a name="line.300"></a>
<FONT color="green">301</FONT>         * @see #getD()<a name="line.301"></a>
<FONT color="green">302</FONT>         * @see #getRealEigenvalues()<a name="line.302"></a>
<FONT color="green">303</FONT>         * @see #getImagEigenvalue(int)<a name="line.303"></a>
<FONT color="green">304</FONT>         */<a name="line.304"></a>
<FONT color="green">305</FONT>        public double getRealEigenvalue(final int i) {<a name="line.305"></a>
<FONT color="green">306</FONT>            return realEigenvalues[i];<a name="line.306"></a>
<FONT color="green">307</FONT>        }<a name="line.307"></a>
<FONT color="green">308</FONT>    <a name="line.308"></a>
<FONT color="green">309</FONT>        /**<a name="line.309"></a>
<FONT color="green">310</FONT>         * Gets a copy of the imaginary parts of the eigenvalues of the original<a name="line.310"></a>
<FONT color="green">311</FONT>         * matrix.<a name="line.311"></a>
<FONT color="green">312</FONT>         *<a name="line.312"></a>
<FONT color="green">313</FONT>         * @return a copy of the imaginary parts of the eigenvalues of the original<a name="line.313"></a>
<FONT color="green">314</FONT>         * matrix.<a name="line.314"></a>
<FONT color="green">315</FONT>         *<a name="line.315"></a>
<FONT color="green">316</FONT>         * @see #getD()<a name="line.316"></a>
<FONT color="green">317</FONT>         * @see #getImagEigenvalue(int)<a name="line.317"></a>
<FONT color="green">318</FONT>         * @see #getRealEigenvalues()<a name="line.318"></a>
<FONT color="green">319</FONT>         */<a name="line.319"></a>
<FONT color="green">320</FONT>        public double[] getImagEigenvalues() {<a name="line.320"></a>
<FONT color="green">321</FONT>            return imagEigenvalues.clone();<a name="line.321"></a>
<FONT color="green">322</FONT>        }<a name="line.322"></a>
<FONT color="green">323</FONT>    <a name="line.323"></a>
<FONT color="green">324</FONT>        /**<a name="line.324"></a>
<FONT color="green">325</FONT>         * Gets the imaginary part of the i&lt;sup&gt;th&lt;/sup&gt; eigenvalue of the original<a name="line.325"></a>
<FONT color="green">326</FONT>         * matrix.<a name="line.326"></a>
<FONT color="green">327</FONT>         *<a name="line.327"></a>
<FONT color="green">328</FONT>         * @param i Index of the eigenvalue (counting from 0).<a name="line.328"></a>
<FONT color="green">329</FONT>         * @return the imaginary part of the i&lt;sup&gt;th&lt;/sup&gt; eigenvalue of the original<a name="line.329"></a>
<FONT color="green">330</FONT>         * matrix.<a name="line.330"></a>
<FONT color="green">331</FONT>         *<a name="line.331"></a>
<FONT color="green">332</FONT>         * @see #getD()<a name="line.332"></a>
<FONT color="green">333</FONT>         * @see #getImagEigenvalues()<a name="line.333"></a>
<FONT color="green">334</FONT>         * @see #getRealEigenvalue(int)<a name="line.334"></a>
<FONT color="green">335</FONT>         */<a name="line.335"></a>
<FONT color="green">336</FONT>        public double getImagEigenvalue(final int i) {<a name="line.336"></a>
<FONT color="green">337</FONT>            return imagEigenvalues[i];<a name="line.337"></a>
<FONT color="green">338</FONT>        }<a name="line.338"></a>
<FONT color="green">339</FONT>    <a name="line.339"></a>
<FONT color="green">340</FONT>        /**<a name="line.340"></a>
<FONT color="green">341</FONT>         * Gets a copy of the i&lt;sup&gt;th&lt;/sup&gt; eigenvector of the original matrix.<a name="line.341"></a>
<FONT color="green">342</FONT>         *<a name="line.342"></a>
<FONT color="green">343</FONT>         * @param i Index of the eigenvector (counting from 0).<a name="line.343"></a>
<FONT color="green">344</FONT>         * @return a copy of the i&lt;sup&gt;th&lt;/sup&gt; eigenvector of the original matrix.<a name="line.344"></a>
<FONT color="green">345</FONT>         * @see #getD()<a name="line.345"></a>
<FONT color="green">346</FONT>         */<a name="line.346"></a>
<FONT color="green">347</FONT>        public RealVector getEigenvector(final int i) {<a name="line.347"></a>
<FONT color="green">348</FONT>            return eigenvectors[i].copy();<a name="line.348"></a>
<FONT color="green">349</FONT>        }<a name="line.349"></a>
<FONT color="green">350</FONT>    <a name="line.350"></a>
<FONT color="green">351</FONT>        /**<a name="line.351"></a>
<FONT color="green">352</FONT>         * Computes the determinant of the matrix.<a name="line.352"></a>
<FONT color="green">353</FONT>         *<a name="line.353"></a>
<FONT color="green">354</FONT>         * @return the determinant of the matrix.<a name="line.354"></a>
<FONT color="green">355</FONT>         */<a name="line.355"></a>
<FONT color="green">356</FONT>        public double getDeterminant() {<a name="line.356"></a>
<FONT color="green">357</FONT>            double determinant = 1;<a name="line.357"></a>
<FONT color="green">358</FONT>            for (double lambda : realEigenvalues) {<a name="line.358"></a>
<FONT color="green">359</FONT>                determinant *= lambda;<a name="line.359"></a>
<FONT color="green">360</FONT>            }<a name="line.360"></a>
<FONT color="green">361</FONT>            return determinant;<a name="line.361"></a>
<FONT color="green">362</FONT>        }<a name="line.362"></a>
<FONT color="green">363</FONT>    <a name="line.363"></a>
<FONT color="green">364</FONT>        /**<a name="line.364"></a>
<FONT color="green">365</FONT>         * Computes the square-root of the matrix.<a name="line.365"></a>
<FONT color="green">366</FONT>         * This implementation assumes that the matrix is symmetric and postive<a name="line.366"></a>
<FONT color="green">367</FONT>         * definite.<a name="line.367"></a>
<FONT color="green">368</FONT>         *<a name="line.368"></a>
<FONT color="green">369</FONT>         * @return the square-root of the matrix.<a name="line.369"></a>
<FONT color="green">370</FONT>         * @throws MathUnsupportedOperationException if the matrix is not<a name="line.370"></a>
<FONT color="green">371</FONT>         * symmetric or not positive definite.<a name="line.371"></a>
<FONT color="green">372</FONT>         * @since 3.1<a name="line.372"></a>
<FONT color="green">373</FONT>         */<a name="line.373"></a>
<FONT color="green">374</FONT>        public RealMatrix getSquareRoot() {<a name="line.374"></a>
<FONT color="green">375</FONT>            if (!isSymmetric) {<a name="line.375"></a>
<FONT color="green">376</FONT>                throw new MathUnsupportedOperationException();<a name="line.376"></a>
<FONT color="green">377</FONT>            }<a name="line.377"></a>
<FONT color="green">378</FONT>    <a name="line.378"></a>
<FONT color="green">379</FONT>            final double[] sqrtEigenValues = new double[realEigenvalues.length];<a name="line.379"></a>
<FONT color="green">380</FONT>            for (int i = 0; i &lt; realEigenvalues.length; i++) {<a name="line.380"></a>
<FONT color="green">381</FONT>                final double eigen = realEigenvalues[i];<a name="line.381"></a>
<FONT color="green">382</FONT>                if (eigen &lt;= 0) {<a name="line.382"></a>
<FONT color="green">383</FONT>                    throw new MathUnsupportedOperationException();<a name="line.383"></a>
<FONT color="green">384</FONT>                }<a name="line.384"></a>
<FONT color="green">385</FONT>                sqrtEigenValues[i] = FastMath.sqrt(eigen);<a name="line.385"></a>
<FONT color="green">386</FONT>            }<a name="line.386"></a>
<FONT color="green">387</FONT>            final RealMatrix sqrtEigen = MatrixUtils.createRealDiagonalMatrix(sqrtEigenValues);<a name="line.387"></a>
<FONT color="green">388</FONT>            final RealMatrix v = getV();<a name="line.388"></a>
<FONT color="green">389</FONT>            final RealMatrix vT = getVT();<a name="line.389"></a>
<FONT color="green">390</FONT>    <a name="line.390"></a>
<FONT color="green">391</FONT>            return v.multiply(sqrtEigen).multiply(vT);<a name="line.391"></a>
<FONT color="green">392</FONT>        }<a name="line.392"></a>
<FONT color="green">393</FONT>    <a name="line.393"></a>
<FONT color="green">394</FONT>        /**<a name="line.394"></a>
<FONT color="green">395</FONT>         * Gets a solver for finding the A &amp;times; X = B solution in exact<a name="line.395"></a>
<FONT color="green">396</FONT>         * linear sense.<a name="line.396"></a>
<FONT color="green">397</FONT>         * &lt;p&gt;<a name="line.397"></a>
<FONT color="green">398</FONT>         * Since 3.1, eigen decomposition of a general matrix is supported,<a name="line.398"></a>
<FONT color="green">399</FONT>         * but the {@link DecompositionSolver} only supports real eigenvalues.<a name="line.399"></a>
<FONT color="green">400</FONT>         *<a name="line.400"></a>
<FONT color="green">401</FONT>         * @return a solver<a name="line.401"></a>
<FONT color="green">402</FONT>         * @throws MathUnsupportedOperationException if the decomposition resulted in<a name="line.402"></a>
<FONT color="green">403</FONT>         * complex eigenvalues<a name="line.403"></a>
<FONT color="green">404</FONT>         */<a name="line.404"></a>
<FONT color="green">405</FONT>        public DecompositionSolver getSolver() {<a name="line.405"></a>
<FONT color="green">406</FONT>            if (hasComplexEigenvalues()) {<a name="line.406"></a>
<FONT color="green">407</FONT>                throw new MathUnsupportedOperationException();<a name="line.407"></a>
<FONT color="green">408</FONT>            }<a name="line.408"></a>
<FONT color="green">409</FONT>            return new Solver(realEigenvalues, imagEigenvalues, eigenvectors);<a name="line.409"></a>
<FONT color="green">410</FONT>        }<a name="line.410"></a>
<FONT color="green">411</FONT>    <a name="line.411"></a>
<FONT color="green">412</FONT>        /** Specialized solver. */<a name="line.412"></a>
<FONT color="green">413</FONT>        private static class Solver implements DecompositionSolver {<a name="line.413"></a>
<FONT color="green">414</FONT>            /** Real part of the realEigenvalues. */<a name="line.414"></a>
<FONT color="green">415</FONT>            private double[] realEigenvalues;<a name="line.415"></a>
<FONT color="green">416</FONT>            /** Imaginary part of the realEigenvalues. */<a name="line.416"></a>
<FONT color="green">417</FONT>            private double[] imagEigenvalues;<a name="line.417"></a>
<FONT color="green">418</FONT>            /** Eigenvectors. */<a name="line.418"></a>
<FONT color="green">419</FONT>            private final ArrayRealVector[] eigenvectors;<a name="line.419"></a>
<FONT color="green">420</FONT>    <a name="line.420"></a>
<FONT color="green">421</FONT>            /**<a name="line.421"></a>
<FONT color="green">422</FONT>             * Builds a solver from decomposed matrix.<a name="line.422"></a>
<FONT color="green">423</FONT>             *<a name="line.423"></a>
<FONT color="green">424</FONT>             * @param realEigenvalues Real parts of the eigenvalues.<a name="line.424"></a>
<FONT color="green">425</FONT>             * @param imagEigenvalues Imaginary parts of the eigenvalues.<a name="line.425"></a>
<FONT color="green">426</FONT>             * @param eigenvectors Eigenvectors.<a name="line.426"></a>
<FONT color="green">427</FONT>             */<a name="line.427"></a>
<FONT color="green">428</FONT>            private Solver(final double[] realEigenvalues,<a name="line.428"></a>
<FONT color="green">429</FONT>                    final double[] imagEigenvalues,<a name="line.429"></a>
<FONT color="green">430</FONT>                    final ArrayRealVector[] eigenvectors) {<a name="line.430"></a>
<FONT color="green">431</FONT>                this.realEigenvalues = realEigenvalues;<a name="line.431"></a>
<FONT color="green">432</FONT>                this.imagEigenvalues = imagEigenvalues;<a name="line.432"></a>
<FONT color="green">433</FONT>                this.eigenvectors = eigenvectors;<a name="line.433"></a>
<FONT color="green">434</FONT>            }<a name="line.434"></a>
<FONT color="green">435</FONT>    <a name="line.435"></a>
<FONT color="green">436</FONT>            /**<a name="line.436"></a>
<FONT color="green">437</FONT>             * Solves the linear equation A &amp;times; X = B for symmetric matrices A.<a name="line.437"></a>
<FONT color="green">438</FONT>             * &lt;p&gt;<a name="line.438"></a>
<FONT color="green">439</FONT>             * This method only finds exact linear solutions, i.e. solutions for<a name="line.439"></a>
<FONT color="green">440</FONT>             * which ||A &amp;times; X - B|| is exactly 0.<a name="line.440"></a>
<FONT color="green">441</FONT>             * &lt;/p&gt;<a name="line.441"></a>
<FONT color="green">442</FONT>             *<a name="line.442"></a>
<FONT color="green">443</FONT>             * @param b Right-hand side of the equation A &amp;times; X = B.<a name="line.443"></a>
<FONT color="green">444</FONT>             * @return a Vector X that minimizes the two norm of A &amp;times; X - B.<a name="line.444"></a>
<FONT color="green">445</FONT>             *<a name="line.445"></a>
<FONT color="green">446</FONT>             * @throws DimensionMismatchException if the matrices dimensions do not match.<a name="line.446"></a>
<FONT color="green">447</FONT>             * @throws SingularMatrixException if the decomposed matrix is singular.<a name="line.447"></a>
<FONT color="green">448</FONT>             */<a name="line.448"></a>
<FONT color="green">449</FONT>            public RealVector solve(final RealVector b) {<a name="line.449"></a>
<FONT color="green">450</FONT>                if (!isNonSingular()) {<a name="line.450"></a>
<FONT color="green">451</FONT>                    throw new SingularMatrixException();<a name="line.451"></a>
<FONT color="green">452</FONT>                }<a name="line.452"></a>
<FONT color="green">453</FONT>    <a name="line.453"></a>
<FONT color="green">454</FONT>                final int m = realEigenvalues.length;<a name="line.454"></a>
<FONT color="green">455</FONT>                if (b.getDimension() != m) {<a name="line.455"></a>
<FONT color="green">456</FONT>                    throw new DimensionMismatchException(b.getDimension(), m);<a name="line.456"></a>
<FONT color="green">457</FONT>                }<a name="line.457"></a>
<FONT color="green">458</FONT>    <a name="line.458"></a>
<FONT color="green">459</FONT>                final double[] bp = new double[m];<a name="line.459"></a>
<FONT color="green">460</FONT>                for (int i = 0; i &lt; m; ++i) {<a name="line.460"></a>
<FONT color="green">461</FONT>                    final ArrayRealVector v = eigenvectors[i];<a name="line.461"></a>
<FONT color="green">462</FONT>                    final double[] vData = v.getDataRef();<a name="line.462"></a>
<FONT color="green">463</FONT>                    final double s = v.dotProduct(b) / realEigenvalues[i];<a name="line.463"></a>
<FONT color="green">464</FONT>                    for (int j = 0; j &lt; m; ++j) {<a name="line.464"></a>
<FONT color="green">465</FONT>                        bp[j] += s * vData[j];<a name="line.465"></a>
<FONT color="green">466</FONT>                    }<a name="line.466"></a>
<FONT color="green">467</FONT>                }<a name="line.467"></a>
<FONT color="green">468</FONT>    <a name="line.468"></a>
<FONT color="green">469</FONT>                return new ArrayRealVector(bp, false);<a name="line.469"></a>
<FONT color="green">470</FONT>            }<a name="line.470"></a>
<FONT color="green">471</FONT>    <a name="line.471"></a>
<FONT color="green">472</FONT>            /** {@inheritDoc} */<a name="line.472"></a>
<FONT color="green">473</FONT>            public RealMatrix solve(RealMatrix b) {<a name="line.473"></a>
<FONT color="green">474</FONT>    <a name="line.474"></a>
<FONT color="green">475</FONT>                if (!isNonSingular()) {<a name="line.475"></a>
<FONT color="green">476</FONT>                    throw new SingularMatrixException();<a name="line.476"></a>
<FONT color="green">477</FONT>                }<a name="line.477"></a>
<FONT color="green">478</FONT>    <a name="line.478"></a>
<FONT color="green">479</FONT>                final int m = realEigenvalues.length;<a name="line.479"></a>
<FONT color="green">480</FONT>                if (b.getRowDimension() != m) {<a name="line.480"></a>
<FONT color="green">481</FONT>                    throw new DimensionMismatchException(b.getRowDimension(), m);<a name="line.481"></a>
<FONT color="green">482</FONT>                }<a name="line.482"></a>
<FONT color="green">483</FONT>    <a name="line.483"></a>
<FONT color="green">484</FONT>                final int nColB = b.getColumnDimension();<a name="line.484"></a>
<FONT color="green">485</FONT>                final double[][] bp = new double[m][nColB];<a name="line.485"></a>
<FONT color="green">486</FONT>                final double[] tmpCol = new double[m];<a name="line.486"></a>
<FONT color="green">487</FONT>                for (int k = 0; k &lt; nColB; ++k) {<a name="line.487"></a>
<FONT color="green">488</FONT>                    for (int i = 0; i &lt; m; ++i) {<a name="line.488"></a>
<FONT color="green">489</FONT>                        tmpCol[i] = b.getEntry(i, k);<a name="line.489"></a>
<FONT color="green">490</FONT>                        bp[i][k]  = 0;<a name="line.490"></a>
<FONT color="green">491</FONT>                    }<a name="line.491"></a>
<FONT color="green">492</FONT>                    for (int i = 0; i &lt; m; ++i) {<a name="line.492"></a>
<FONT color="green">493</FONT>                        final ArrayRealVector v = eigenvectors[i];<a name="line.493"></a>
<FONT color="green">494</FONT>                        final double[] vData = v.getDataRef();<a name="line.494"></a>
<FONT color="green">495</FONT>                        double s = 0;<a name="line.495"></a>
<FONT color="green">496</FONT>                        for (int j = 0; j &lt; m; ++j) {<a name="line.496"></a>
<FONT color="green">497</FONT>                            s += v.getEntry(j) * tmpCol[j];<a name="line.497"></a>
<FONT color="green">498</FONT>                        }<a name="line.498"></a>
<FONT color="green">499</FONT>                        s /= realEigenvalues[i];<a name="line.499"></a>
<FONT color="green">500</FONT>                        for (int j = 0; j &lt; m; ++j) {<a name="line.500"></a>
<FONT color="green">501</FONT>                            bp[j][k] += s * vData[j];<a name="line.501"></a>
<FONT color="green">502</FONT>                        }<a name="line.502"></a>
<FONT color="green">503</FONT>                    }<a name="line.503"></a>
<FONT color="green">504</FONT>                }<a name="line.504"></a>
<FONT color="green">505</FONT>    <a name="line.505"></a>
<FONT color="green">506</FONT>                return new Array2DRowRealMatrix(bp, false);<a name="line.506"></a>
<FONT color="green">507</FONT>    <a name="line.507"></a>
<FONT color="green">508</FONT>            }<a name="line.508"></a>
<FONT color="green">509</FONT>    <a name="line.509"></a>
<FONT color="green">510</FONT>            /**<a name="line.510"></a>
<FONT color="green">511</FONT>             * Checks whether the decomposed matrix is non-singular.<a name="line.511"></a>
<FONT color="green">512</FONT>             *<a name="line.512"></a>
<FONT color="green">513</FONT>             * @return true if the decomposed matrix is non-singular.<a name="line.513"></a>
<FONT color="green">514</FONT>             */<a name="line.514"></a>
<FONT color="green">515</FONT>            public boolean isNonSingular() {<a name="line.515"></a>
<FONT color="green">516</FONT>                for (int i = 0; i &lt; realEigenvalues.length; ++i) {<a name="line.516"></a>
<FONT color="green">517</FONT>                    if (realEigenvalues[i] == 0 &amp;&amp;<a name="line.517"></a>
<FONT color="green">518</FONT>                        imagEigenvalues[i] == 0) {<a name="line.518"></a>
<FONT color="green">519</FONT>                        return false;<a name="line.519"></a>
<FONT color="green">520</FONT>                    }<a name="line.520"></a>
<FONT color="green">521</FONT>                }<a name="line.521"></a>
<FONT color="green">522</FONT>                return true;<a name="line.522"></a>
<FONT color="green">523</FONT>            }<a name="line.523"></a>
<FONT color="green">524</FONT>    <a name="line.524"></a>
<FONT color="green">525</FONT>            /**<a name="line.525"></a>
<FONT color="green">526</FONT>             * Get the inverse of the decomposed matrix.<a name="line.526"></a>
<FONT color="green">527</FONT>             *<a name="line.527"></a>
<FONT color="green">528</FONT>             * @return the inverse matrix.<a name="line.528"></a>
<FONT color="green">529</FONT>             * @throws SingularMatrixException if the decomposed matrix is singular.<a name="line.529"></a>
<FONT color="green">530</FONT>             */<a name="line.530"></a>
<FONT color="green">531</FONT>            public RealMatrix getInverse() {<a name="line.531"></a>
<FONT color="green">532</FONT>                if (!isNonSingular()) {<a name="line.532"></a>
<FONT color="green">533</FONT>                    throw new SingularMatrixException();<a name="line.533"></a>
<FONT color="green">534</FONT>                }<a name="line.534"></a>
<FONT color="green">535</FONT>    <a name="line.535"></a>
<FONT color="green">536</FONT>                final int m = realEigenvalues.length;<a name="line.536"></a>
<FONT color="green">537</FONT>                final double[][] invData = new double[m][m];<a name="line.537"></a>
<FONT color="green">538</FONT>    <a name="line.538"></a>
<FONT color="green">539</FONT>                for (int i = 0; i &lt; m; ++i) {<a name="line.539"></a>
<FONT color="green">540</FONT>                    final double[] invI = invData[i];<a name="line.540"></a>
<FONT color="green">541</FONT>                    for (int j = 0; j &lt; m; ++j) {<a name="line.541"></a>
<FONT color="green">542</FONT>                        double invIJ = 0;<a name="line.542"></a>
<FONT color="green">543</FONT>                        for (int k = 0; k &lt; m; ++k) {<a name="line.543"></a>
<FONT color="green">544</FONT>                            final double[] vK = eigenvectors[k].getDataRef();<a name="line.544"></a>
<FONT color="green">545</FONT>                            invIJ += vK[i] * vK[j] / realEigenvalues[k];<a name="line.545"></a>
<FONT color="green">546</FONT>                        }<a name="line.546"></a>
<FONT color="green">547</FONT>                        invI[j] = invIJ;<a name="line.547"></a>
<FONT color="green">548</FONT>                    }<a name="line.548"></a>
<FONT color="green">549</FONT>                }<a name="line.549"></a>
<FONT color="green">550</FONT>                return MatrixUtils.createRealMatrix(invData);<a name="line.550"></a>
<FONT color="green">551</FONT>            }<a name="line.551"></a>
<FONT color="green">552</FONT>        }<a name="line.552"></a>
<FONT color="green">553</FONT>    <a name="line.553"></a>
<FONT color="green">554</FONT>        /**<a name="line.554"></a>
<FONT color="green">555</FONT>         * Transforms the matrix to tridiagonal form.<a name="line.555"></a>
<FONT color="green">556</FONT>         *<a name="line.556"></a>
<FONT color="green">557</FONT>         * @param matrix Matrix to transform.<a name="line.557"></a>
<FONT color="green">558</FONT>         */<a name="line.558"></a>
<FONT color="green">559</FONT>        private void transformToTridiagonal(final RealMatrix matrix) {<a name="line.559"></a>
<FONT color="green">560</FONT>            // transform the matrix to tridiagonal<a name="line.560"></a>
<FONT color="green">561</FONT>            transformer = new TriDiagonalTransformer(matrix);<a name="line.561"></a>
<FONT color="green">562</FONT>            main = transformer.getMainDiagonalRef();<a name="line.562"></a>
<FONT color="green">563</FONT>            secondary = transformer.getSecondaryDiagonalRef();<a name="line.563"></a>
<FONT color="green">564</FONT>        }<a name="line.564"></a>
<FONT color="green">565</FONT>    <a name="line.565"></a>
<FONT color="green">566</FONT>        /**<a name="line.566"></a>
<FONT color="green">567</FONT>         * Find eigenvalues and eigenvectors (Dubrulle et al., 1971)<a name="line.567"></a>
<FONT color="green">568</FONT>         *<a name="line.568"></a>
<FONT color="green">569</FONT>         * @param householderMatrix Householder matrix of the transformation<a name="line.569"></a>
<FONT color="green">570</FONT>         * to tridiagonal form.<a name="line.570"></a>
<FONT color="green">571</FONT>         */<a name="line.571"></a>
<FONT color="green">572</FONT>        private void findEigenVectors(final double[][] householderMatrix) {<a name="line.572"></a>
<FONT color="green">573</FONT>            final double[][]z = householderMatrix.clone();<a name="line.573"></a>
<FONT color="green">574</FONT>            final int n = main.length;<a name="line.574"></a>
<FONT color="green">575</FONT>            realEigenvalues = new double[n];<a name="line.575"></a>
<FONT color="green">576</FONT>            imagEigenvalues = new double[n];<a name="line.576"></a>
<FONT color="green">577</FONT>            final double[] e = new double[n];<a name="line.577"></a>
<FONT color="green">578</FONT>            for (int i = 0; i &lt; n - 1; i++) {<a name="line.578"></a>
<FONT color="green">579</FONT>                realEigenvalues[i] = main[i];<a name="line.579"></a>
<FONT color="green">580</FONT>                e[i] = secondary[i];<a name="line.580"></a>
<FONT color="green">581</FONT>            }<a name="line.581"></a>
<FONT color="green">582</FONT>            realEigenvalues[n - 1] = main[n - 1];<a name="line.582"></a>
<FONT color="green">583</FONT>            e[n - 1] = 0;<a name="line.583"></a>
<FONT color="green">584</FONT>    <a name="line.584"></a>
<FONT color="green">585</FONT>            // Determine the largest main and secondary value in absolute term.<a name="line.585"></a>
<FONT color="green">586</FONT>            double maxAbsoluteValue = 0;<a name="line.586"></a>
<FONT color="green">587</FONT>            for (int i = 0; i &lt; n; i++) {<a name="line.587"></a>
<FONT color="green">588</FONT>                if (FastMath.abs(realEigenvalues[i]) &gt; maxAbsoluteValue) {<a name="line.588"></a>
<FONT color="green">589</FONT>                    maxAbsoluteValue = FastMath.abs(realEigenvalues[i]);<a name="line.589"></a>
<FONT color="green">590</FONT>                }<a name="line.590"></a>
<FONT color="green">591</FONT>                if (FastMath.abs(e[i]) &gt; maxAbsoluteValue) {<a name="line.591"></a>
<FONT color="green">592</FONT>                    maxAbsoluteValue = FastMath.abs(e[i]);<a name="line.592"></a>
<FONT color="green">593</FONT>                }<a name="line.593"></a>
<FONT color="green">594</FONT>            }<a name="line.594"></a>
<FONT color="green">595</FONT>            // Make null any main and secondary value too small to be significant<a name="line.595"></a>
<FONT color="green">596</FONT>            if (maxAbsoluteValue != 0) {<a name="line.596"></a>
<FONT color="green">597</FONT>                for (int i=0; i &lt; n; i++) {<a name="line.597"></a>
<FONT color="green">598</FONT>                    if (FastMath.abs(realEigenvalues[i]) &lt;= Precision.EPSILON * maxAbsoluteValue) {<a name="line.598"></a>
<FONT color="green">599</FONT>                        realEigenvalues[i] = 0;<a name="line.599"></a>
<FONT color="green">600</FONT>                    }<a name="line.600"></a>
<FONT color="green">601</FONT>                    if (FastMath.abs(e[i]) &lt;= Precision.EPSILON * maxAbsoluteValue) {<a name="line.601"></a>
<FONT color="green">602</FONT>                        e[i]=0;<a name="line.602"></a>
<FONT color="green">603</FONT>                    }<a name="line.603"></a>
<FONT color="green">604</FONT>                }<a name="line.604"></a>
<FONT color="green">605</FONT>            }<a name="line.605"></a>
<FONT color="green">606</FONT>    <a name="line.606"></a>
<FONT color="green">607</FONT>            for (int j = 0; j &lt; n; j++) {<a name="line.607"></a>
<FONT color="green">608</FONT>                int its = 0;<a name="line.608"></a>
<FONT color="green">609</FONT>                int m;<a name="line.609"></a>
<FONT color="green">610</FONT>                do {<a name="line.610"></a>
<FONT color="green">611</FONT>                    for (m = j; m &lt; n - 1; m++) {<a name="line.611"></a>
<FONT color="green">612</FONT>                        double delta = FastMath.abs(realEigenvalues[m]) +<a name="line.612"></a>
<FONT color="green">613</FONT>                            FastMath.abs(realEigenvalues[m + 1]);<a name="line.613"></a>
<FONT color="green">614</FONT>                        if (FastMath.abs(e[m]) + delta == delta) {<a name="line.614"></a>
<FONT color="green">615</FONT>                            break;<a name="line.615"></a>
<FONT color="green">616</FONT>                        }<a name="line.616"></a>
<FONT color="green">617</FONT>                    }<a name="line.617"></a>
<FONT color="green">618</FONT>                    if (m != j) {<a name="line.618"></a>
<FONT color="green">619</FONT>                        if (its == maxIter) {<a name="line.619"></a>
<FONT color="green">620</FONT>                            throw new MaxCountExceededException(LocalizedFormats.CONVERGENCE_FAILED,<a name="line.620"></a>
<FONT color="green">621</FONT>                                                                maxIter);<a name="line.621"></a>
<FONT color="green">622</FONT>                        }<a name="line.622"></a>
<FONT color="green">623</FONT>                        its++;<a name="line.623"></a>
<FONT color="green">624</FONT>                        double q = (realEigenvalues[j + 1] - realEigenvalues[j]) / (2 * e[j]);<a name="line.624"></a>
<FONT color="green">625</FONT>                        double t = FastMath.sqrt(1 + q * q);<a name="line.625"></a>
<FONT color="green">626</FONT>                        if (q &lt; 0.0) {<a name="line.626"></a>
<FONT color="green">627</FONT>                            q = realEigenvalues[m] - realEigenvalues[j] + e[j] / (q - t);<a name="line.627"></a>
<FONT color="green">628</FONT>                        } else {<a name="line.628"></a>
<FONT color="green">629</FONT>                            q = realEigenvalues[m] - realEigenvalues[j] + e[j] / (q + t);<a name="line.629"></a>
<FONT color="green">630</FONT>                        }<a name="line.630"></a>
<FONT color="green">631</FONT>                        double u = 0.0;<a name="line.631"></a>
<FONT color="green">632</FONT>                        double s = 1.0;<a name="line.632"></a>
<FONT color="green">633</FONT>                        double c = 1.0;<a name="line.633"></a>
<FONT color="green">634</FONT>                        int i;<a name="line.634"></a>
<FONT color="green">635</FONT>                        for (i = m - 1; i &gt;= j; i--) {<a name="line.635"></a>
<FONT color="green">636</FONT>                            double p = s * e[i];<a name="line.636"></a>
<FONT color="green">637</FONT>                            double h = c * e[i];<a name="line.637"></a>
<FONT color="green">638</FONT>                            if (FastMath.abs(p) &gt;= FastMath.abs(q)) {<a name="line.638"></a>
<FONT color="green">639</FONT>                                c = q / p;<a name="line.639"></a>
<FONT color="green">640</FONT>                                t = FastMath.sqrt(c * c + 1.0);<a name="line.640"></a>
<FONT color="green">641</FONT>                                e[i + 1] = p * t;<a name="line.641"></a>
<FONT color="green">642</FONT>                                s = 1.0 / t;<a name="line.642"></a>
<FONT color="green">643</FONT>                                c = c * s;<a name="line.643"></a>
<FONT color="green">644</FONT>                            } else {<a name="line.644"></a>
<FONT color="green">645</FONT>                                s = p / q;<a name="line.645"></a>
<FONT color="green">646</FONT>                                t = FastMath.sqrt(s * s + 1.0);<a name="line.646"></a>
<FONT color="green">647</FONT>                                e[i + 1] = q * t;<a name="line.647"></a>
<FONT color="green">648</FONT>                                c = 1.0 / t;<a name="line.648"></a>
<FONT color="green">649</FONT>                                s = s * c;<a name="line.649"></a>
<FONT color="green">650</FONT>                            }<a name="line.650"></a>
<FONT color="green">651</FONT>                            if (e[i + 1] == 0.0) {<a name="line.651"></a>
<FONT color="green">652</FONT>                                realEigenvalues[i + 1] -= u;<a name="line.652"></a>
<FONT color="green">653</FONT>                                e[m] = 0.0;<a name="line.653"></a>
<FONT color="green">654</FONT>                                break;<a name="line.654"></a>
<FONT color="green">655</FONT>                            }<a name="line.655"></a>
<FONT color="green">656</FONT>                            q = realEigenvalues[i + 1] - u;<a name="line.656"></a>
<FONT color="green">657</FONT>                            t = (realEigenvalues[i] - q) * s + 2.0 * c * h;<a name="line.657"></a>
<FONT color="green">658</FONT>                            u = s * t;<a name="line.658"></a>
<FONT color="green">659</FONT>                            realEigenvalues[i + 1] = q + u;<a name="line.659"></a>
<FONT color="green">660</FONT>                            q = c * t - h;<a name="line.660"></a>
<FONT color="green">661</FONT>                            for (int ia = 0; ia &lt; n; ia++) {<a name="line.661"></a>
<FONT color="green">662</FONT>                                p = z[ia][i + 1];<a name="line.662"></a>
<FONT color="green">663</FONT>                                z[ia][i + 1] = s * z[ia][i] + c * p;<a name="line.663"></a>
<FONT color="green">664</FONT>                                z[ia][i] = c * z[ia][i] - s * p;<a name="line.664"></a>
<FONT color="green">665</FONT>                            }<a name="line.665"></a>
<FONT color="green">666</FONT>                        }<a name="line.666"></a>
<FONT color="green">667</FONT>                        if (t == 0.0 &amp;&amp; i &gt;= j) {<a name="line.667"></a>
<FONT color="green">668</FONT>                            continue;<a name="line.668"></a>
<FONT color="green">669</FONT>                        }<a name="line.669"></a>
<FONT color="green">670</FONT>                        realEigenvalues[j] -= u;<a name="line.670"></a>
<FONT color="green">671</FONT>                        e[j] = q;<a name="line.671"></a>
<FONT color="green">672</FONT>                        e[m] = 0.0;<a name="line.672"></a>
<FONT color="green">673</FONT>                    }<a name="line.673"></a>
<FONT color="green">674</FONT>                } while (m != j);<a name="line.674"></a>
<FONT color="green">675</FONT>            }<a name="line.675"></a>
<FONT color="green">676</FONT>    <a name="line.676"></a>
<FONT color="green">677</FONT>            //Sort the eigen values (and vectors) in increase order<a name="line.677"></a>
<FONT color="green">678</FONT>            for (int i = 0; i &lt; n; i++) {<a name="line.678"></a>
<FONT color="green">679</FONT>                int k = i;<a name="line.679"></a>
<FONT color="green">680</FONT>                double p = realEigenvalues[i];<a name="line.680"></a>
<FONT color="green">681</FONT>                for (int j = i + 1; j &lt; n; j++) {<a name="line.681"></a>
<FONT color="green">682</FONT>                    if (realEigenvalues[j] &gt; p) {<a name="line.682"></a>
<FONT color="green">683</FONT>                        k = j;<a name="line.683"></a>
<FONT color="green">684</FONT>                        p = realEigenvalues[j];<a name="line.684"></a>
<FONT color="green">685</FONT>                    }<a name="line.685"></a>
<FONT color="green">686</FONT>                }<a name="line.686"></a>
<FONT color="green">687</FONT>                if (k != i) {<a name="line.687"></a>
<FONT color="green">688</FONT>                    realEigenvalues[k] = realEigenvalues[i];<a name="line.688"></a>
<FONT color="green">689</FONT>                    realEigenvalues[i] = p;<a name="line.689"></a>
<FONT color="green">690</FONT>                    for (int j = 0; j &lt; n; j++) {<a name="line.690"></a>
<FONT color="green">691</FONT>                        p = z[j][i];<a name="line.691"></a>
<FONT color="green">692</FONT>                        z[j][i] = z[j][k];<a name="line.692"></a>
<FONT color="green">693</FONT>                        z[j][k] = p;<a name="line.693"></a>
<FONT color="green">694</FONT>                    }<a name="line.694"></a>
<FONT color="green">695</FONT>                }<a name="line.695"></a>
<FONT color="green">696</FONT>            }<a name="line.696"></a>
<FONT color="green">697</FONT>    <a name="line.697"></a>
<FONT color="green">698</FONT>            // Determine the largest eigen value in absolute term.<a name="line.698"></a>
<FONT color="green">699</FONT>            maxAbsoluteValue = 0;<a name="line.699"></a>
<FONT color="green">700</FONT>            for (int i = 0; i &lt; n; i++) {<a name="line.700"></a>
<FONT color="green">701</FONT>                if (FastMath.abs(realEigenvalues[i]) &gt; maxAbsoluteValue) {<a name="line.701"></a>
<FONT color="green">702</FONT>                    maxAbsoluteValue=FastMath.abs(realEigenvalues[i]);<a name="line.702"></a>
<FONT color="green">703</FONT>                }<a name="line.703"></a>
<FONT color="green">704</FONT>            }<a name="line.704"></a>
<FONT color="green">705</FONT>            // Make null any eigen value too small to be significant<a name="line.705"></a>
<FONT color="green">706</FONT>            if (maxAbsoluteValue != 0.0) {<a name="line.706"></a>
<FONT color="green">707</FONT>                for (int i=0; i &lt; n; i++) {<a name="line.707"></a>
<FONT color="green">708</FONT>                    if (FastMath.abs(realEigenvalues[i]) &lt; Precision.EPSILON * maxAbsoluteValue) {<a name="line.708"></a>
<FONT color="green">709</FONT>                        realEigenvalues[i] = 0;<a name="line.709"></a>
<FONT color="green">710</FONT>                    }<a name="line.710"></a>
<FONT color="green">711</FONT>                }<a name="line.711"></a>
<FONT color="green">712</FONT>            }<a name="line.712"></a>
<FONT color="green">713</FONT>            eigenvectors = new ArrayRealVector[n];<a name="line.713"></a>
<FONT color="green">714</FONT>            final double[] tmp = new double[n];<a name="line.714"></a>
<FONT color="green">715</FONT>            for (int i = 0; i &lt; n; i++) {<a name="line.715"></a>
<FONT color="green">716</FONT>                for (int j = 0; j &lt; n; j++) {<a name="line.716"></a>
<FONT color="green">717</FONT>                    tmp[j] = z[j][i];<a name="line.717"></a>
<FONT color="green">718</FONT>                }<a name="line.718"></a>
<FONT color="green">719</FONT>                eigenvectors[i] = new ArrayRealVector(tmp);<a name="line.719"></a>
<FONT color="green">720</FONT>            }<a name="line.720"></a>
<FONT color="green">721</FONT>        }<a name="line.721"></a>
<FONT color="green">722</FONT>    <a name="line.722"></a>
<FONT color="green">723</FONT>        /**<a name="line.723"></a>
<FONT color="green">724</FONT>         * Transforms the matrix to Schur form and calculates the eigenvalues.<a name="line.724"></a>
<FONT color="green">725</FONT>         *<a name="line.725"></a>
<FONT color="green">726</FONT>         * @param matrix Matrix to transform.<a name="line.726"></a>
<FONT color="green">727</FONT>         * @return the {@link SchurTransformer Shur transform} for this matrix<a name="line.727"></a>
<FONT color="green">728</FONT>         */<a name="line.728"></a>
<FONT color="green">729</FONT>        private SchurTransformer transformToSchur(final RealMatrix matrix) {<a name="line.729"></a>
<FONT color="green">730</FONT>            final SchurTransformer schurTransform = new SchurTransformer(matrix);<a name="line.730"></a>
<FONT color="green">731</FONT>            final double[][] matT = schurTransform.getT().getData();<a name="line.731"></a>
<FONT color="green">732</FONT>    <a name="line.732"></a>
<FONT color="green">733</FONT>            realEigenvalues = new double[matT.length];<a name="line.733"></a>
<FONT color="green">734</FONT>            imagEigenvalues = new double[matT.length];<a name="line.734"></a>
<FONT color="green">735</FONT>    <a name="line.735"></a>
<FONT color="green">736</FONT>            for (int i = 0; i &lt; realEigenvalues.length; i++) {<a name="line.736"></a>
<FONT color="green">737</FONT>                if (i == (realEigenvalues.length - 1) ||<a name="line.737"></a>
<FONT color="green">738</FONT>                    Precision.equals(matT[i + 1][i], 0.0, EPSILON)) {<a name="line.738"></a>
<FONT color="green">739</FONT>                    realEigenvalues[i] = matT[i][i];<a name="line.739"></a>
<FONT color="green">740</FONT>                } else {<a name="line.740"></a>
<FONT color="green">741</FONT>                    final double x = matT[i + 1][i + 1];<a name="line.741"></a>
<FONT color="green">742</FONT>                    final double p = 0.5 * (matT[i][i] - x);<a name="line.742"></a>
<FONT color="green">743</FONT>                    final double z = FastMath.sqrt(FastMath.abs(p * p + matT[i + 1][i] * matT[i][i + 1]));<a name="line.743"></a>
<FONT color="green">744</FONT>                    realEigenvalues[i] = x + p;<a name="line.744"></a>
<FONT color="green">745</FONT>                    imagEigenvalues[i] = z;<a name="line.745"></a>
<FONT color="green">746</FONT>                    realEigenvalues[i + 1] = x + p;<a name="line.746"></a>
<FONT color="green">747</FONT>                    imagEigenvalues[i + 1] = -z;<a name="line.747"></a>
<FONT color="green">748</FONT>                    i++;<a name="line.748"></a>
<FONT color="green">749</FONT>                }<a name="line.749"></a>
<FONT color="green">750</FONT>            }<a name="line.750"></a>
<FONT color="green">751</FONT>            return schurTransform;<a name="line.751"></a>
<FONT color="green">752</FONT>        }<a name="line.752"></a>
<FONT color="green">753</FONT>    <a name="line.753"></a>
<FONT color="green">754</FONT>        /**<a name="line.754"></a>
<FONT color="green">755</FONT>         * Performs a division of two complex numbers.<a name="line.755"></a>
<FONT color="green">756</FONT>         *<a name="line.756"></a>
<FONT color="green">757</FONT>         * @param xr real part of the first number<a name="line.757"></a>
<FONT color="green">758</FONT>         * @param xi imaginary part of the first number<a name="line.758"></a>
<FONT color="green">759</FONT>         * @param yr real part of the second number<a name="line.759"></a>
<FONT color="green">760</FONT>         * @param yi imaginary part of the second number<a name="line.760"></a>
<FONT color="green">761</FONT>         * @return result of the complex division<a name="line.761"></a>
<FONT color="green">762</FONT>         */<a name="line.762"></a>
<FONT color="green">763</FONT>        private Complex cdiv(final double xr, final double xi,<a name="line.763"></a>
<FONT color="green">764</FONT>                             final double yr, final double yi) {<a name="line.764"></a>
<FONT color="green">765</FONT>            return new Complex(xr, xi).divide(new Complex(yr, yi));<a name="line.765"></a>
<FONT color="green">766</FONT>        }<a name="line.766"></a>
<FONT color="green">767</FONT>    <a name="line.767"></a>
<FONT color="green">768</FONT>        /**<a name="line.768"></a>
<FONT color="green">769</FONT>         * Find eigenvectors from a matrix transformed to Schur form.<a name="line.769"></a>
<FONT color="green">770</FONT>         *<a name="line.770"></a>
<FONT color="green">771</FONT>         * @param schur the schur transformation of the matrix<a name="line.771"></a>
<FONT color="green">772</FONT>         * @throws MathArithmeticException if the Schur form has a norm of zero<a name="line.772"></a>
<FONT color="green">773</FONT>         */<a name="line.773"></a>
<FONT color="green">774</FONT>        private void findEigenVectorsFromSchur(final SchurTransformer schur)<a name="line.774"></a>
<FONT color="green">775</FONT>            throws MathArithmeticException {<a name="line.775"></a>
<FONT color="green">776</FONT>            final double[][] matrixT = schur.getT().getData();<a name="line.776"></a>
<FONT color="green">777</FONT>            final double[][] matrixP = schur.getP().getData();<a name="line.777"></a>
<FONT color="green">778</FONT>    <a name="line.778"></a>
<FONT color="green">779</FONT>            final int n = matrixT.length;<a name="line.779"></a>
<FONT color="green">780</FONT>    <a name="line.780"></a>
<FONT color="green">781</FONT>            // compute matrix norm<a name="line.781"></a>
<FONT color="green">782</FONT>            double norm = 0.0;<a name="line.782"></a>
<FONT color="green">783</FONT>            for (int i = 0; i &lt; n; i++) {<a name="line.783"></a>
<FONT color="green">784</FONT>               for (int j = FastMath.max(i - 1, 0); j &lt; n; j++) {<a name="line.784"></a>
<FONT color="green">785</FONT>                  norm = norm + FastMath.abs(matrixT[i][j]);<a name="line.785"></a>
<FONT color="green">786</FONT>               }<a name="line.786"></a>
<FONT color="green">787</FONT>            }<a name="line.787"></a>
<FONT color="green">788</FONT>    <a name="line.788"></a>
<FONT color="green">789</FONT>            // we can not handle a matrix with zero norm<a name="line.789"></a>
<FONT color="green">790</FONT>            if (Precision.equals(norm, 0.0, EPSILON)) {<a name="line.790"></a>
<FONT color="green">791</FONT>               throw new MathArithmeticException(LocalizedFormats.ZERO_NORM);<a name="line.791"></a>
<FONT color="green">792</FONT>            }<a name="line.792"></a>
<FONT color="green">793</FONT>    <a name="line.793"></a>
<FONT color="green">794</FONT>            // Backsubstitute to find vectors of upper triangular form<a name="line.794"></a>
<FONT color="green">795</FONT>    <a name="line.795"></a>
<FONT color="green">796</FONT>            double r = 0.0;<a name="line.796"></a>
<FONT color="green">797</FONT>            double s = 0.0;<a name="line.797"></a>
<FONT color="green">798</FONT>            double z = 0.0;<a name="line.798"></a>
<FONT color="green">799</FONT>    <a name="line.799"></a>
<FONT color="green">800</FONT>            for (int idx = n - 1; idx &gt;= 0; idx--) {<a name="line.800"></a>
<FONT color="green">801</FONT>                double p = realEigenvalues[idx];<a name="line.801"></a>
<FONT color="green">802</FONT>                double q = imagEigenvalues[idx];<a name="line.802"></a>
<FONT color="green">803</FONT>    <a name="line.803"></a>
<FONT color="green">804</FONT>                if (Precision.equals(q, 0.0)) {<a name="line.804"></a>
<FONT color="green">805</FONT>                    // Real vector<a name="line.805"></a>
<FONT color="green">806</FONT>                    int l = idx;<a name="line.806"></a>
<FONT color="green">807</FONT>                    matrixT[idx][idx] = 1.0;<a name="line.807"></a>
<FONT color="green">808</FONT>                    for (int i = idx - 1; i &gt;= 0; i--) {<a name="line.808"></a>
<FONT color="green">809</FONT>                        double w = matrixT[i][i] - p;<a name="line.809"></a>
<FONT color="green">810</FONT>                        r = 0.0;<a name="line.810"></a>
<FONT color="green">811</FONT>                        for (int j = l; j &lt;= idx; j++) {<a name="line.811"></a>
<FONT color="green">812</FONT>                            r = r + matrixT[i][j] * matrixT[j][idx];<a name="line.812"></a>
<FONT color="green">813</FONT>                        }<a name="line.813"></a>
<FONT color="green">814</FONT>                        if (Precision.compareTo(imagEigenvalues[i], 0.0, EPSILON) &lt; 0.0) {<a name="line.814"></a>
<FONT color="green">815</FONT>                            z = w;<a name="line.815"></a>
<FONT color="green">816</FONT>                            s = r;<a name="line.816"></a>
<FONT color="green">817</FONT>                        } else {<a name="line.817"></a>
<FONT color="green">818</FONT>                            l = i;<a name="line.818"></a>
<FONT color="green">819</FONT>                            if (Precision.equals(imagEigenvalues[i], 0.0)) {<a name="line.819"></a>
<FONT color="green">820</FONT>                                if (w != 0.0) {<a name="line.820"></a>
<FONT color="green">821</FONT>                                    matrixT[i][idx] = -r / w;<a name="line.821"></a>
<FONT color="green">822</FONT>                                } else {<a name="line.822"></a>
<FONT color="green">823</FONT>                                    matrixT[i][idx] = -r / (Precision.EPSILON * norm);<a name="line.823"></a>
<FONT color="green">824</FONT>                                }<a name="line.824"></a>
<FONT color="green">825</FONT>                            } else {<a name="line.825"></a>
<FONT color="green">826</FONT>                                // Solve real equations<a name="line.826"></a>
<FONT color="green">827</FONT>                                double x = matrixT[i][i + 1];<a name="line.827"></a>
<FONT color="green">828</FONT>                                double y = matrixT[i + 1][i];<a name="line.828"></a>
<FONT color="green">829</FONT>                                q = (realEigenvalues[i] - p) * (realEigenvalues[i] - p) +<a name="line.829"></a>
<FONT color="green">830</FONT>                                    imagEigenvalues[i] * imagEigenvalues[i];<a name="line.830"></a>
<FONT color="green">831</FONT>                                double t = (x * s - z * r) / q;<a name="line.831"></a>
<FONT color="green">832</FONT>                                matrixT[i][idx] = t;<a name="line.832"></a>
<FONT color="green">833</FONT>                                if (FastMath.abs(x) &gt; FastMath.abs(z)) {<a name="line.833"></a>
<FONT color="green">834</FONT>                                    matrixT[i + 1][idx] = (-r - w * t) / x;<a name="line.834"></a>
<FONT color="green">835</FONT>                                } else {<a name="line.835"></a>
<FONT color="green">836</FONT>                                    matrixT[i + 1][idx] = (-s - y * t) / z;<a name="line.836"></a>
<FONT color="green">837</FONT>                                }<a name="line.837"></a>
<FONT color="green">838</FONT>                            }<a name="line.838"></a>
<FONT color="green">839</FONT>    <a name="line.839"></a>
<FONT color="green">840</FONT>                            // Overflow control<a name="line.840"></a>
<FONT color="green">841</FONT>                            double t = FastMath.abs(matrixT[i][idx]);<a name="line.841"></a>
<FONT color="green">842</FONT>                            if ((Precision.EPSILON * t) * t &gt; 1) {<a name="line.842"></a>
<FONT color="green">843</FONT>                                for (int j = i; j &lt;= idx; j++) {<a name="line.843"></a>
<FONT color="green">844</FONT>                                    matrixT[j][idx] = matrixT[j][idx] / t;<a name="line.844"></a>
<FONT color="green">845</FONT>                                }<a name="line.845"></a>
<FONT color="green">846</FONT>                            }<a name="line.846"></a>
<FONT color="green">847</FONT>                        }<a name="line.847"></a>
<FONT color="green">848</FONT>                    }<a name="line.848"></a>
<FONT color="green">849</FONT>                } else if (q &lt; 0.0) {<a name="line.849"></a>
<FONT color="green">850</FONT>                    // Complex vector<a name="line.850"></a>
<FONT color="green">851</FONT>                    int l = idx - 1;<a name="line.851"></a>
<FONT color="green">852</FONT>    <a name="line.852"></a>
<FONT color="green">853</FONT>                    // Last vector component imaginary so matrix is triangular<a name="line.853"></a>
<FONT color="green">854</FONT>                    if (FastMath.abs(matrixT[idx][idx - 1]) &gt; FastMath.abs(matrixT[idx - 1][idx])) {<a name="line.854"></a>
<FONT color="green">855</FONT>                        matrixT[idx - 1][idx - 1] = q / matrixT[idx][idx - 1];<a name="line.855"></a>
<FONT color="green">856</FONT>                        matrixT[idx - 1][idx]     = -(matrixT[idx][idx] - p) / matrixT[idx][idx - 1];<a name="line.856"></a>
<FONT color="green">857</FONT>                    } else {<a name="line.857"></a>
<FONT color="green">858</FONT>                        final Complex result = cdiv(0.0, -matrixT[idx - 1][idx],<a name="line.858"></a>
<FONT color="green">859</FONT>                                                    matrixT[idx - 1][idx - 1] - p, q);<a name="line.859"></a>
<FONT color="green">860</FONT>                        matrixT[idx - 1][idx - 1] = result.getReal();<a name="line.860"></a>
<FONT color="green">861</FONT>                        matrixT[idx - 1][idx]     = result.getImaginary();<a name="line.861"></a>
<FONT color="green">862</FONT>                    }<a name="line.862"></a>
<FONT color="green">863</FONT>    <a name="line.863"></a>
<FONT color="green">864</FONT>                    matrixT[idx][idx - 1] = 0.0;<a name="line.864"></a>
<FONT color="green">865</FONT>                    matrixT[idx][idx]     = 1.0;<a name="line.865"></a>
<FONT color="green">866</FONT>    <a name="line.866"></a>
<FONT color="green">867</FONT>                    for (int i = idx - 2; i &gt;= 0; i--) {<a name="line.867"></a>
<FONT color="green">868</FONT>                        double ra = 0.0;<a name="line.868"></a>
<FONT color="green">869</FONT>                        double sa = 0.0;<a name="line.869"></a>
<FONT color="green">870</FONT>                        for (int j = l; j &lt;= idx; j++) {<a name="line.870"></a>
<FONT color="green">871</FONT>                            ra = ra + matrixT[i][j] * matrixT[j][idx - 1];<a name="line.871"></a>
<FONT color="green">872</FONT>                            sa = sa + matrixT[i][j] * matrixT[j][idx];<a name="line.872"></a>
<FONT color="green">873</FONT>                        }<a name="line.873"></a>
<FONT color="green">874</FONT>                        double w = matrixT[i][i] - p;<a name="line.874"></a>
<FONT color="green">875</FONT>    <a name="line.875"></a>
<FONT color="green">876</FONT>                        if (Precision.compareTo(imagEigenvalues[i], 0.0, EPSILON) &lt; 0.0) {<a name="line.876"></a>
<FONT color="green">877</FONT>                            z = w;<a name="line.877"></a>
<FONT color="green">878</FONT>                            r = ra;<a name="line.878"></a>
<FONT color="green">879</FONT>                            s = sa;<a name="line.879"></a>
<FONT color="green">880</FONT>                        } else {<a name="line.880"></a>
<FONT color="green">881</FONT>                            l = i;<a name="line.881"></a>
<FONT color="green">882</FONT>                            if (Precision.equals(imagEigenvalues[i], 0.0)) {<a name="line.882"></a>
<FONT color="green">883</FONT>                                final Complex c = cdiv(-ra, -sa, w, q);<a name="line.883"></a>
<FONT color="green">884</FONT>                                matrixT[i][idx - 1] = c.getReal();<a name="line.884"></a>
<FONT color="green">885</FONT>                                matrixT[i][idx] = c.getImaginary();<a name="line.885"></a>
<FONT color="green">886</FONT>                            } else {<a name="line.886"></a>
<FONT color="green">887</FONT>                                // Solve complex equations<a name="line.887"></a>
<FONT color="green">888</FONT>                                double x = matrixT[i][i + 1];<a name="line.888"></a>
<FONT color="green">889</FONT>                                double y = matrixT[i + 1][i];<a name="line.889"></a>
<FONT color="green">890</FONT>                                double vr = (realEigenvalues[i] - p) * (realEigenvalues[i] - p) +<a name="line.890"></a>
<FONT color="green">891</FONT>                                            imagEigenvalues[i] * imagEigenvalues[i] - q * q;<a name="line.891"></a>
<FONT color="green">892</FONT>                                final double vi = (realEigenvalues[i] - p) * 2.0 * q;<a name="line.892"></a>
<FONT color="green">893</FONT>                                if (Precision.equals(vr, 0.0) &amp;&amp; Precision.equals(vi, 0.0)) {<a name="line.893"></a>
<FONT color="green">894</FONT>                                    vr = Precision.EPSILON * norm *<a name="line.894"></a>
<FONT color="green">895</FONT>                                         (FastMath.abs(w) + FastMath.abs(q) + FastMath.abs(x) +<a name="line.895"></a>
<FONT color="green">896</FONT>                                          FastMath.abs(y) + FastMath.abs(z));<a name="line.896"></a>
<FONT color="green">897</FONT>                                }<a name="line.897"></a>
<FONT color="green">898</FONT>                                final Complex c     = cdiv(x * r - z * ra + q * sa,<a name="line.898"></a>
<FONT color="green">899</FONT>                                                           x * s - z * sa - q * ra, vr, vi);<a name="line.899"></a>
<FONT color="green">900</FONT>                                matrixT[i][idx - 1] = c.getReal();<a name="line.900"></a>
<FONT color="green">901</FONT>                                matrixT[i][idx]     = c.getImaginary();<a name="line.901"></a>
<FONT color="green">902</FONT>    <a name="line.902"></a>
<FONT color="green">903</FONT>                                if (FastMath.abs(x) &gt; (FastMath.abs(z) + FastMath.abs(q))) {<a name="line.903"></a>
<FONT color="green">904</FONT>                                    matrixT[i + 1][idx - 1] = (-ra - w * matrixT[i][idx - 1] +<a name="line.904"></a>
<FONT color="green">905</FONT>                                                               q * matrixT[i][idx]) / x;<a name="line.905"></a>
<FONT color="green">906</FONT>                                    matrixT[i + 1][idx]     = (-sa - w * matrixT[i][idx] -<a name="line.906"></a>
<FONT color="green">907</FONT>                                                               q * matrixT[i][idx - 1]) / x;<a name="line.907"></a>
<FONT color="green">908</FONT>                                } else {<a name="line.908"></a>
<FONT color="green">909</FONT>                                    final Complex c2        = cdiv(-r - y * matrixT[i][idx - 1],<a name="line.909"></a>
<FONT color="green">910</FONT>                                                                   -s - y * matrixT[i][idx], z, q);<a name="line.910"></a>
<FONT color="green">911</FONT>                                    matrixT[i + 1][idx - 1] = c2.getReal();<a name="line.911"></a>
<FONT color="green">912</FONT>                                    matrixT[i + 1][idx]     = c2.getImaginary();<a name="line.912"></a>
<FONT color="green">913</FONT>                                }<a name="line.913"></a>
<FONT color="green">914</FONT>                            }<a name="line.914"></a>
<FONT color="green">915</FONT>    <a name="line.915"></a>
<FONT color="green">916</FONT>                            // Overflow control<a name="line.916"></a>
<FONT color="green">917</FONT>                            double t = FastMath.max(FastMath.abs(matrixT[i][idx - 1]),<a name="line.917"></a>
<FONT color="green">918</FONT>                                                    FastMath.abs(matrixT[i][idx]));<a name="line.918"></a>
<FONT color="green">919</FONT>                            if ((Precision.EPSILON * t) * t &gt; 1) {<a name="line.919"></a>
<FONT color="green">920</FONT>                                for (int j = i; j &lt;= idx; j++) {<a name="line.920"></a>
<FONT color="green">921</FONT>                                    matrixT[j][idx - 1] = matrixT[j][idx - 1] / t;<a name="line.921"></a>
<FONT color="green">922</FONT>                                    matrixT[j][idx]     = matrixT[j][idx] / t;<a name="line.922"></a>
<FONT color="green">923</FONT>                                }<a name="line.923"></a>
<FONT color="green">924</FONT>                            }<a name="line.924"></a>
<FONT color="green">925</FONT>                        }<a name="line.925"></a>
<FONT color="green">926</FONT>                    }<a name="line.926"></a>
<FONT color="green">927</FONT>                }<a name="line.927"></a>
<FONT color="green">928</FONT>            }<a name="line.928"></a>
<FONT color="green">929</FONT>    <a name="line.929"></a>
<FONT color="green">930</FONT>            // Vectors of isolated roots<a name="line.930"></a>
<FONT color="green">931</FONT>            for (int i = 0; i &lt; n; i++) {<a name="line.931"></a>
<FONT color="green">932</FONT>                if (i &lt; 0 | i &gt; n - 1) {<a name="line.932"></a>
<FONT color="green">933</FONT>                    for (int j = i; j &lt; n; j++) {<a name="line.933"></a>
<FONT color="green">934</FONT>                        matrixP[i][j] = matrixT[i][j];<a name="line.934"></a>
<FONT color="green">935</FONT>                    }<a name="line.935"></a>
<FONT color="green">936</FONT>                }<a name="line.936"></a>
<FONT color="green">937</FONT>            }<a name="line.937"></a>
<FONT color="green">938</FONT>    <a name="line.938"></a>
<FONT color="green">939</FONT>            // Back transformation to get eigenvectors of original matrix<a name="line.939"></a>
<FONT color="green">940</FONT>            for (int j = n - 1; j &gt;= 0; j--) {<a name="line.940"></a>
<FONT color="green">941</FONT>                for (int i = 0; i &lt;= n - 1; i++) {<a name="line.941"></a>
<FONT color="green">942</FONT>                    z = 0.0;<a name="line.942"></a>
<FONT color="green">943</FONT>                    for (int k = 0; k &lt;= FastMath.min(j, n - 1); k++) {<a name="line.943"></a>
<FONT color="green">944</FONT>                        z = z + matrixP[i][k] * matrixT[k][j];<a name="line.944"></a>
<FONT color="green">945</FONT>                    }<a name="line.945"></a>
<FONT color="green">946</FONT>                    matrixP[i][j] = z;<a name="line.946"></a>
<FONT color="green">947</FONT>                }<a name="line.947"></a>
<FONT color="green">948</FONT>            }<a name="line.948"></a>
<FONT color="green">949</FONT>    <a name="line.949"></a>
<FONT color="green">950</FONT>            eigenvectors = new ArrayRealVector[n];<a name="line.950"></a>
<FONT color="green">951</FONT>            final double[] tmp = new double[n];<a name="line.951"></a>
<FONT color="green">952</FONT>            for (int i = 0; i &lt; n; i++) {<a name="line.952"></a>
<FONT color="green">953</FONT>                for (int j = 0; j &lt; n; j++) {<a name="line.953"></a>
<FONT color="green">954</FONT>                    tmp[j] = matrixP[j][i];<a name="line.954"></a>
<FONT color="green">955</FONT>                }<a name="line.955"></a>
<FONT color="green">956</FONT>                eigenvectors[i] = new ArrayRealVector(tmp);<a name="line.956"></a>
<FONT color="green">957</FONT>            }<a name="line.957"></a>
<FONT color="green">958</FONT>        }<a name="line.958"></a>
<FONT color="green">959</FONT>    }<a name="line.959"></a>




























































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